Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/463

When the original variables are by direct substitution replaced by the new variables in the differential equations, the latter take the form

${\begin{matrix}{\text{div}}_{1}\ \mathbf {d} _{1}&=&4\pi c^{2}\rho _{1},\qquad \qquad &{\text{curl}}_{1}\ \mathbf {d} _{1}&=&-\partial \mathbf {h} _{1}/\partial t_{1},&\ \\{\text{div}}_{1}\ \mathbf {h} _{1}&=&0,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {h} _{1}&=&(1/c^{2})\partial \mathbf {d} _{1}/\partial t_{1}&+\ 4\pi \rho _{1}\mathbf {v} _{1},\end{matrix}}$

that is to say, the fundamental equations of the aether retain their form unaltered, when the variables are subjected to the transformation which has been specified.

We are now in a position to show the connexion of this transformation with FitzGerald's hypothesis of contraction. Suppose that two material particles are moving along the axis of x with velocity w = c tanh α. From the relation

$v_{x}=c\tanh \alpha +{\frac {v_{x_{1}}{\text{sech }}\alpha }{\cosh \alpha +v_{x_{1}}c^{-1}\sinh \alpha }}$,

it follows that $v_{x_{1}}$, is zero for each of the particles, which implies that they are at rest relative to the new axes. Let x₁, and x′₁ denote their coordinates with respect to this latter system; then the coordinates of one particle at the instant t₁, referred to the original axes, will be given by the equations

$x=x_{1}\cosh \alpha +ct_{1}\sinh \alpha ,\qquad t=t_{1}\cosh \alpha +x_{1}c^{-1}\sinh \alpha$;

and the coordinates of the other particle will be given by

$x^{\prime }=x_{1}^{\prime }\cosh \alpha +ct_{1}\sinh \alpha ,\qquad t^{\prime }=t_{1}\cosh \alpha +x_{1}^{\prime }c^{-1}\sinh \alpha$;

so that at time t the latter particle will have the coordinate x′′, where

${\begin{matrix}x^{\prime \prime }&=&x^{\prime }+w(t-t^{\prime })\\\ &=&x_{1}^{\prime }\cosh \alpha +ct_{1}\sinh \alpha +(x-x_{1}^{\prime })\sinh ^{2}\alpha {\text{ sech }}\alpha ,\end{matrix}}$

which gives

$x^{\prime \prime }-x=(x_{1}^{\prime }-x_{1})(1-w^{2}/c^{2})^{\frac {1}{2}}$.

This equation shows that the distance between the particles in the system of measurement furnished by the original axes, with reference to which the particles were moving with velocity w, bears the ratio (1 - w²/c²)^1/2:1 to their distance in the